In a paper written in collaboration with R. McCann, we prove existence of solution of a relativistic heat equation via the Jordan-Kinderlehere-Otto scheme.

This method is strongly based on the existence of an optimal map for the Monge-Kantorovich problem with a relativistic cost.

In the study of the Jordan-Kinderlehrer-Otto scheme, we obtain this optimal map using a double minimization process but the general problem of the existence of an optimal map for such a cost is still open.

I will also present this problem which is the heart of a new collaboration with Jerome Bertrand still in progress.